Engineering Mathematics Volume - II

833 Pages·2015·12.306 MB·English

by  R. L. Garg, Nishu Gupta

Engineering Mathematics
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Engineering Mathematics
Vol. II
R. L. Garg
Nishu Gupta
Delhi (cid:31) Chennai
No part of this eBook may be used or reproduced in any manner whatsoever without the
publisher’s prior written consent.
Copyright © 2015 Published by Dorling Kindersley (India) Pvt. Ltd
This eBook may or may not include all assets that were part of the print version. The publisher
reserves the right to remove any material in this eBook at any time.
ISBN: 9789332536333
e-ISBN: 9789332542181
First Impression
Head Office: 7th Floor, Knowledge Boulevard, A-8(A), Sector 62, Noida 201 309, UP, India.
Registered Office: 11 Community Centre, Panchsheel Park, New Delhi 110 017, India.
Dedication
To My Wife
Late Smt. Shashi Kiran Garg
R. L. Garg
To My Parents
Smt. Adarsh Garg
and
Late Shri B.S.Garg
Nishu Gupta
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Contents
Preface xiii
Acknowledgements xiv
AbouttheAuthors xv
Symbols,BasicFormulaeandUsefulInformations xvii
1.1 Introduction — 1
1.2 Definition, Limit, Continuity and Differentiability of a Function of Complex Variable — 1
1.2.1Definition 1
1.2.2LimitofaFunction 1
1.2.3ContinuityofaFunction 4
1.2.4DifferentiabilityofaFunction 4
1.3 Analytic Functions — 5
1.4 Cauchy–Riemann Equations — 5
1.4.1SufficientConditionsforaFunctiontobeAnalytic 6
1.4.2PolarFormofCauchy–RiemannEquations 7
1.5 Harmonic Functions — 21
1.5.1OrthogonalSystemofLevelCurves 22
1.5.2MethodtoFindConjugateHarmonicFunction 23
1.5.3MilneThomsonMethod 23
1.6 Line Integral in the Complex Plane — 39
1.6.1ContinuousCurveorArc 39
1.6.2SmoothCurveorSmoothArc 39
1.6.3PiecewiseContinuousCurve 39
1.6.4PiecewiseSmoothCurve 39
1.6.5Contour 39
1.6.6LineIntegral 39
1.7 Cauchy Integral Theorem — 44
1.7.1SimplyConnectedDomain 44
1.7.2MultiplyConnectedDomain 44
1.7.3IndependenceofPath 46
1.7.4IntegralFunction 47
1.7.5FundamentalTheoremofIntegralCalculus 47
1.7.6ExtensionofCauchy–GoursatTheoremforMultiplyConnectedDomains 49
viii | Contents
1.8 Cauchy Integral Formula — 50
1.8.1CauchyIntegralFormulaforDerivativesofAnalyticFunction 51
1.8.2Morera’sTheorem(ConverseofCauchyIntegralTheorem) 53
1.8.3CauchyInequality 53
1.8.4Liouville’sTheorem 54
1.8.5Poisson’sIntegralFormula 54
1.9 Infinite Series of Complex Terms — 66
1.9.1PowerSeries 67
1.10 Taylor Series — 68
1.11 Laurent’s Series — 69
1.12 Zeros and Singularities of Complex Functions — 84
1.12.1ZerosofanAnalyticFunction 84
1.12.2SingularitiesofaFunction 84
1.12.3MethodtoFindTypeofIsolatedSingularity 85
1.13 Residue — 90
1.13.1ResidueataRemovableSingularity 90
1.13.2ResidueataSimplePole 90
1.13.3ResidueatPoleofOrderm 91
1.13.4ResidueatanIsolatedEssentialSingularity 91
1.14 Evaluation of Contour Integrals using Residues — 97
1.15 Application of Cauchy Residue Theorem to Evaluate Real Integrals — 106
1.15.1IntegrationAroundtheUnitCircle 106
∞ ∞
1.15.2ImproperRealIntegralsoftheForm ∫f(x)dxor∫f(x)dxwheref(z)hasno
RealSingularity 114 −∞ 0
1.15.3SomeSpecialImproperRealIntegrals 121
1.15.4ImproperIntegralswithSingularitiesonRealAxis 122
1.16 Conformal Mapping — 132
1.17 Some Standard Mappings — 136
1.17.1TranslationMapping 136
1.17.2Magnification/ContractionandRotation 136
1.17.3LinearTransformation 137
1.17.4InverseTransformation(InversionandReflection) 139
1.17.5SquareTransformation 144
1.17.6BilinearTransformation(MobiusTransformationorFractionalTransformation) 149
1.17.7CrossRatioofFourPoints 149
2.1 Introduction — 165
2.2 Definition of Laplace Transform and Inverse Laplace Transform — 165
2.2.1PiecewiseContinuousFunction 166
2.2.2FunctionofExponentialOrder 166
Contents | ix
2.3 Sufficient Conditions for Existence of Laplace Transform — 166
2.4 Properties of Laplace Transforms — 167
2.5 Laplace Transform of Elementary Functions — 168
2.6 Laplace Transforms of Derivatives and Integrals — 170
2.7 Differentiation and Integration of Laplace Transform — 172
2.8 Evaluation of Real Integrals using Laplace Transform — 187
2.9 Laplace Transform of Unit Step Function — 192
2.10 Laplace Transform of Unit Impulse Function (Dirac–Delta Function) — 199
2.11 Laplace Transform of Periodic Functions — 202
2.12 Inverse Laplace Transform — 209
2.13 Use of Partial Fractions to Find Inverse Laplace Transform — 210
2.14 Convolution Theorem — 221
2.15 Applications of Laplace Transform to Solve Linear Differential Equations,
Simultaneous Linear Differential Equations and Integral Equations — 227
2.16 Applications of Laplace Transform to Engineering Problems — 257
2.16.1ProblemsRelatedtoElectricalCircuits 257
2.16.2ProblemRelatedtoDeflectionofaLoadedBeam 265
2.16.3ProblemsRelatedtoMechanicalSystems 271
3.1 Introduction — 281
3.1.1PeriodicFunctions 281
3.1.2TrigonometricSeries 282
3.1.3OrthogonalityofTrigonometricSystem 282
3.1.4FourierSeries 283
3.1.5EulerFormulaeforFourierCoefficients 283
3.1.6Dirichlet’sConditionsforConvergenceofFourierSeriesoff(x)in[c,c+2l] 286
3.1.7FourierSeriesofEvenandOddFunctions 286
3.2 Fourier Half-range Series — 323
3.2.1ConvergenceofHalf-rangeCosineSeries 324
3.2.2ConvergenceofHalf-rangeSineSeries 324
3.3 Others Formulae — 341
3.3.1Parseval’sFormulae 341
3.3.2RootMeanSquare(R.M.S.)Value 343
3.3.3ComplexFormofFourierSeries 343
3.4 Harmonic Analysis — 352
3.5 Fourier Integrals and Fourier Transforms — 367
3.5.1FourierSeriestoFourierIntegrals 367
3.5.2FourierCosineandFourierSineIntegrals 368
3.5.3FourierCosineandSineTransforms 369
3.5.4ComplexFormoftheFourierIntegral 370